(A) Let \(\ell \)  be a length function exponentially distorted on \(N^{\prime }\) . Using Theorem REF  we can
write every \(g\in G\)  as
\(g=\exp (\eta )\exp (\xi )l\) , where \((\eta ,\xi ,l)\in \mathfrak {n}^{\prime }\times \mathfrak {v}\times L\) . Then
\(\ell (\exp (\eta )\exp (\xi )l)\le \ell (\exp (\eta ))+\ell (\exp (\xi ))+\ell (l).\) 
It suffices to estimate every summand on the right by the corresponding summand in (REF ).
First, Lemma REF  implies that \(N^{\prime }\)  is nilpotent, closed and simply connected. So by Part (A) of Proposition REF , we have that \(\ell (\exp (\eta ))\lesssim \log (1+|\eta |)\)  on \(\mathfrak {n}^{\prime }\) .
Second, Lemma REF  implies that \(\ell \lesssim \ell _1\tau \)  on \(\exp \mathfrak {v}\)  (here \(\ell _1\)  is a word length function on \(B/N^{\prime }\) ). To estimate the third summand note that each length function is dominated by a word length function. Applying this to \(L\)  we get \(\ell (l)\lesssim \ell _2(l)\) .
The proof of Part (A) is complete.
(B) Being linear, \(G\)  admits a faithful finite-dimensional holomorphic representation \(\pi \) , which we can treat as a homomorphism to a finite-dimensional Banach algebra. Let \(\sigma \)  denote the quotient homomorphism \(G\rightarrow G/N^{\prime }\)  and \(\tilde{\ell }\)  a symmetric word length function on \(G/N^{\prime }\) . Note that \(\tilde{\ell }\sigma \)  is a length function on \(G\)  and it is exponentially distorted on \(N^{\prime }\)  being equal to 0 on it. Now put
\(\ell ^{\prime }\!:=\tilde{\ell }\sigma +\ell _\pi ^{sym},\) 
where \(\ell _\pi ^{sym}\)  is defined as in (REF ).
Since a finite-dimensional algebra is a PI-algebra (see, e.g., [1]}), it follows from Proposition REF  that \(\ell ^{\prime }\)  is strictly exponentially distorted on \(N^{\prime }\) .
By Part (A), we have \(\ell ^{\prime }\lesssim \varphi \) . To complete the proof we need to show that \(\varphi \lesssim \ell ^{\prime }\) .
First we claim that
\(\ell ^{\prime }(\exp (\eta ))+ \tilde{\ell }\sigma (x) \lesssim \ell ^{\prime }(\exp (\eta )x)\quad \text{on $\mathfrak {n}^{\prime }\times X$,}\) 
where \(X\!:=(\exp \mathfrak {v})L\) . Indeed, since both \(\tilde{\ell }\)  and \(\ell ^{\prime }\)  are symmetric, it follows from [2]} that it suffices to show that
\(\ell ^{\prime }\lesssim \tilde{\ell }\sigma \quad \text{on $X$} \quad \text{and} \quad \tilde{\ell }\sigma \lesssim \ell ^{\prime }\quad \text{on $G$}.\) 
The second condition holds by the definition of \(\ell ^{\prime }\) . To verify the first condition note that by Lemma REF , \(\ell ^{\prime }\lesssim \ell _1\tau \)  on \(\exp \mathfrak {v}\) . On the other hand,
\(\ell ^{\prime }\lesssim \ell _2\)  on \(L\)  because \(\ell _2\)  a word length function (here
\(\ell _1\)  and \(\ell _2\)  are word length functions on \(B/N^{\prime }\)  and \(L\) , respectively). Therefore,
\(\ell ^{\prime }(\exp (\xi )l)\le \ell ^{\prime }(\exp (\xi ))+\ell ^{\prime }(l) \lesssim \ell _1\tau (\exp (\xi ))+\ell _2(l) \quad \text{on $\mathfrak {v}\times L$.}\) 
We have by [2]} that \(G/E\cong B/E\times L\) . Since \(E\subset N^{\prime }\subset B\) , we obtain \(G/N^{\prime }\cong B/N^{\prime }\times L\) . Being a word length function on \(G/N^{\prime }\) , the function \(\tilde{\ell }\)  is equivalent to the sum of word length functions.
In particular,
\(\ell _1\tau (\exp (\xi ))+\ell _2(l)\lesssim \tilde{\ell }\sigma ((\exp (\xi )l)\quad \text{on $\mathfrak {v}\times L$}.\) 
Combining (REF ) and (REF ), we have the first condition in (REF ) and so the claim is proved.
Further, it follows from (REF ) and (REF ) that
\(\ell ^{\prime }(\exp (\eta ))+\ell _1\tau (\exp (\xi ))+\ell _2(l)\lesssim \ell ^{\prime }(\exp (\eta )\exp (\xi )l)\quad \text{on $\mathfrak {n}^{\prime }\times \mathfrak {v}\times L$}.\) 
Since \(\ell ^{\prime }\)  is exponentially distorted on \(N^{\prime }\) , Proposition REF  implies that \(\ell ^{\prime }(\exp (\eta ))\simeq \log (1+|\eta |)\)  on \(\mathfrak {n}^{\prime }\) . Thus \(\varphi \lesssim \ell ^{\prime }\) .
Finally, note that a word length function always separates points. Thus being composition of \(G\rightarrow G/N^{\prime }\)  and a word length function, \(\tilde{\ell }\sigma \)  separates the cosets of \(N^{\prime }\) . Also \(\ell ^{\prime }\)  separates the points of \(N^{\prime }\)  being strictly exponentially distorted on it. Hence \(\ell ^{\prime }\)  separates the points of \(G\) .
